PPrepared for submission to JHEP
Stirring a black hole
Julija Markeviˇci¯ut ˙e, Jorge E. Santos
Department of Applied Mathematics and Theoretical Physics,University of Cambridge,Cambridge, CB3 0WA, UK [email protected] , [email protected] Abstract:
We present novel asymptotically global AdS solutions, constructed by turningon a dipolar differential rotation at the conformal boundary. At fixed energy and bound-ary profile, we find two different geometries: a horizonless spacetime, and a deformed,hourglass shaped black hole with zero net angular momentum. Both solutions exist up tosome maximum amplitudes of the boundary profile, and develop an ergoregion attached tothe boundary before the maximum amplitude is reached. We show that both spacetimesdevelop hair as soon as the ergoregion develops. Furthermore, we discuss the full phasediagram, including the possibility of phases with disconnected horizons, by considering theMathisson-Papapetrou equations for a spinning test particle. Finally, we provide a firstprinciple derivation of the superradiant bound purely from CFT data, and outline possiblescenarios for the late time evolution of the system. a r X i v : . [ h e p - t h ] F e b ontents – 1 – Perturbative expansion 34
The behaviour of quantum field theories (QFTs) on a fixed curved spacetime is largelyunknown territory, posing both conceptual and technical challenges. Perhaps the onlyknown universal phenomenon is the emission of Hawking radiation around asymptoticallyflat black holes, and its concomitant information paradox [1]. However, even in the absenceof horizons, interesting novel phenomenology can arise when considering quantum fieldtheory on curved spacetimes, such as vacuum polarisation or particle production.Much of our current understanding of QFTs on fixed nontrivial backgrounds comesfrom perturbation theory. The reasons for this are twofold: we do not know how to analyt-ically study generic strongly coupled theories and we do not yet have a lattice formulationfor a QFT on a generic curved spacetime. This in turn implies that questions that involvestrongly coupled phenomenology, such as confinement, are out of reach and might bringnovel effects. The gauge/gravity duality [2–5] offers a unique opportunity to study suchscenarios.The gauge/gravity duality maps a class of d − dimensional string theories with antide-Sitter boundary conditions (AdS), to certain non-gravitational conformal quantum fieldtheories (CFTs) formulated in ˜ d dimensions, with ˜ d < d . CFTs typically come with twoparameters: a dimensionless coupling λ which measures the interaction of the microscopicconstituents of the theory, and a measure of its degrees of freedom, which we will denoteby N . One of the realisations in [2] was the fact that in the limit where λ → + ∞ and N → + ∞ , the corresponding string theory duals become classical supergravity theories.This is precisely the regime where the dual CFT becomes strongly coupled, and where weexpect interesting new phenomena to arise.A seemingly unrelated topic is that of superradiance , the study of which was pioneeredby [6–13] in the asymptotically flat context, and the first quantitative study involvingrotating black holes in AdS featured in [14–17]. The idea behind the so called superradiantinstability is simple: rotating black holes in AdS with sufficient angular velocity Ω H canhave ergoregions, which can act as reservoirs for negative energy if waves with azimuthalquantum number m and frequency 0 < ω < m Ω H are scattered. If a rotating black holewith an ergoregion is surrounded by a confining potential (in our case this is provided by theAdS potential, but it could be equally provided by a mass term [11, 18] for asymptoticallyflat spacetimes), then this process will be recurrent and will eventually lead to an instability.The endpoint of this instability seems to be outside the scope of current numerical methodsin AdS, however, in [15, 19, 20] it has been argued that the instability is likely to lead toa pointwise violation of the weak cosmic censorship conjecture in AdS.One might have expected superradiance to produce an interesting phase diagram forCFTs in compact spacetimes, via the gauge/gravity duality. While this is certainly truein the context of the microcanonical ensemble, the grand-canonical ensemble seems rather– 2 – φ Ω ( θ ) θ = π/ φ Ω ( θ ) Figure 1 : Illustration of the novel AdS geometries. Left : Solitonic solutions with adifferentially rotating boundary.
Right : Black hole solutions. The rotational pull force ismaximal at θ = π/
4, deforming the black hole horizon into an hourglass shape.innocuous. The reason for this is that the region of moduli space unstable to superradianceis cloaked by a first order Hawking-Page phase transition [21]. This implies that, for afixed temperature and angular velocity that render the Kerr-AdS black hole unstable, thepreferred phase is that of thermal AdS!We shall see that this is no longer the case if we consider nontrivial boundary metricsthat have differential rotation. For each boundary profile metric, we find two classes of so-lutions: a solitonic horizonless spacetime and black hole solutions, which are schematicallypictured in Fig. 1. The solitonic solutions have no horizon, and correspond to deforma-tions of global AdS , while black hole solutions can be seen as continuous deformations ofSchwarzschild-AdS black holes. We will find that both phases develop hair due to super-radiance if the boundary deformation is sufficiently large. In addition, we shall see thatthe phase diagram of solutions is rather intricate, including certain regions of moduli spacewith up to six-fold degeneracy.One might wonder about the interpretation of our results in terms of CFT data. Weshall see that we will be able to reconstruct the critical value of the amplitude for whichthe system develops hair from pure CFT data. This will allow us to identify which CFTobservables might be sensitive to this phenomenon and to the hypothetical violation ofweak cosmic censorship in the bulk.For simplicity, our work will focus on four-dimensional bulk physics and can easily beembedded into a full string theory embedding, as for instance in ABJM [22]. While someof what we describe might follow through to the higher-dimensional case, there are someimportant differences. For instance, for d >
4, the Gregory-Laflamme instability mightalso play a role, and render some of our solutions unstable before they have a chance ofbecoming superradiant. We note however, that such an instability is also likely to lead toa violation of the weak cosmic censorship conjecture [23–25]. In this sense, we expect therelevant physics to be easier to dissect in d = 4.This paper is organised as follows. Section 3 details the construction of soliton andblack hole solutions. In section 4 we present results, and carry out thermodynamical– 3 –nalysis. Section 5 presents stability analysis in which we consider a massless scalar field.Section 6 discusses the CFT interpretation of our results and attempts to identify thefield theory variables that best describe it. In section 7, we study possible multi horizonequilibrium configurations for spinning test particles. Finally, in section 8, we briefly discussthe holographic dual of theories living on a fixed rotating background geometry containingno ergoregions. We conclude in section 9 with discussion and an outline for further work. We start with the four-dimensional Einstein-Hilbert action supplemented by the Gibbons-Hawking-York boundary term [26, 27]: S = 116 πG N (cid:90) M d x √− g (cid:18) R [ g ] + 6 L (cid:19) + 18 πG N (cid:90) ∂ M d x √− γ K, (2.1)where ∂ M is the boundary of AdS , γ µν is the induced metric on the boundary, K is thetrace of the extrinsic curvature of the boundary, L is the length scale of AdS and R [ g ] theRicci scalar. The equations of motion derived from (2.1) read R µν + 3 L g µν = 0 , and we are looking for asymptotically AdS solutions, with a rotating conformal boundarygiven by d s ∂ = − d t + d θ + sin θ (cid:2) d φ + Ω( θ )d t (cid:3) , (2.2)with a dipolar differential rotation profile Ω( θ ) = ε cos θ . We will be interested in findingsolutions where ∂ t and ∂ φ extend as Killing fields into the bulk. Close to the poles and onthe equator, ∂ t is always timelike for any value of ε . However, its norm (cid:107) ∂ t (cid:107) = − ε θ (2.3)is maximal at θ = π/
4, and ∂ t becomes spacelike for certain regions of θ if ε >
2. We shallsee that both black holes and solitons develop hair due to superradiance if ε >
2. Onemight wonder whether this could have been anticipate using the results of Green, Hollands,Ishibashi and Wald [28]. However, the theorems presented in [28] were deduced under theassumption that the boundary metric was metrically that of the Einstein static universe,which is not the case in our setup. Our results suggest that a generalisation of the resultsin [28] should exist even for nontrivial boundary metrics such as ours.
To solve the Einstein equation we employ the DeTurck method [29, 30] (for a review see [31])where we instead solve the equivalent Einstein-DeTurck equation R µν + 3 L g µν − ∇ ( µ ξ ν ) = 0 , (3.1)– 4 –here ξ µ = g νρ (cid:0) Γ µνρ [ g ] − Γ µνρ [˜ g ] (cid:1) is the DeTurck vector and ˜ g is an appropriate referencemetric of our choice (as detailed in the following subsection). Provided that ξ µ = 0, solu-tions to (3.1) will also be solutions of the Einstein equation. Fortunately, it has recentlybeen shown that, for spacetimes possessing a ( t, φ ) → − ( t, φ ) reflection symmetry, solu-tions with ξ (cid:54) = 0 cannot exist [32] - our spacetimes enjoy such a symmetry. We can thusmonitor the norm of ξ to provide a global measure of our numerical integration scheme(see Appendix A).The DeTuck method has the advantage of rendering the equations elliptic, and con-veniently fixing the gauge. We solve (3.1) using the Newton-Raphson method with pseu-dospectral collocation on a square half–half Chebyshev grid. Because our ansatz is writtenin such a way that all unknown functions are even about two of the grid edges (see thenext subsection), we can halve the Chebyshev grid while effectively interpolating over thefull grid. This in turn allows us to use a smaller grid size to achieve a desired numericalaccuracy. Convergence of our solutions is presented in Appendix A (Fig. 21). In order to find solitonic solutions we consider the following numerical ansatzd s = L (1 − y ) (cid:40) − Q d t + 4 Q d y − y + y (2 − y ) (cid:34) Q − x (cid:18) d x + xy (cid:112) − x Q d y (cid:19) + (1 − x ) Q (cid:16) d φ + yx (cid:112) − x Q d t (cid:17) (cid:35)(cid:41) , (3.2)with Q i , i ∈ { , . . . , } , being functions of ( x, y ). If Q = Q = Q = Q = 1 and Q = Q = 0, the line element (3.2) reduces to that of pure AdS in global coordinates,with y being related to the usual radial coordinate in AdS via r = L y (cid:112) − y / (1 − y ),and x parametrising the standard polar angle on S with the identification sin θ = 1 − x .The powers of y and 1 − x appearing in the ansatz ensure that the metric is regular atthe origin y = 0 and axis of rotation x = ± Q i .The reference metric ˜ g used in the DeTurck method is chosen by setting Q = ε , and Q = Q = Q = Q = Q + 1 = 1, everywhere in the bulk. The parameter ε controls therotation on the boundary and in turn the stirring of the bulk.As we are considering a dipolar problem we have reflection symmetry x → − x and thusboth coordinates take values in the square domain [0 , × [0 , Q i are constructed to be even around x = 0 (the equator) and y = 0 (the origin), we can usea square half–half Chebyshev grid clustering around the x = 1 and y = 1 boundaries andthus effectively interpolate the functions with half the number of points. This is especiallyadvantageous as spectral convergence is exponential with grid size. In addition, it fixessmoothness of the geometry at the reflection boundary.This leaves us with two boundaries, one at the conformal boundary located at y = 1,and one at x = 1. The latter is a physical boundary, whereas the former is a fictitious– 5 –oundary where regularity must be imposed. Expanding the equations of motion near x = 1 as a power series in (1 − x ) gives Q (1 , y ) = Q (1 , y ) and Q (1 , y ) = 0, as well as ∂ x Q = ∂ x Q = ∂ x Q = ∂ x Q = ∂ x Q = 0.At the conformal boundary, we demand the metric (3.2) approaches global AdS witha rotating conformal boundary metric, that is to say, we require Q ( x,
1) = Q ( x,
1) = Q ( x,
1) = Q ( x,
1) = Q ( x,
1) + 1 = 1 , and Q ( x,
1) = ε . (3.3)Comparing the conformal boundary metric obtained from our boundary conditions (3.3)and (2.2), gives Ω( θ ) = ε cos θ , in standard polar coordinates on S . For the black hole solutions we use the ansatzd s = L (1 − y ) (cid:40) − y ˜∆( y ) Q d t + 4 y Q d y ˜∆( y ) + y (cid:34) Q − x (cid:16) d x + x (cid:112) − x y Q d y (cid:17) + (1 − x ) Q (cid:16) d φ + y x (cid:112) − x Q d t (cid:17) (cid:35)(cid:41) , (3.4a)where∆( y ) = (1 − y ) + y (3 − y + y ) , and ˜∆( y ) = ∆( y ) δ + y (1 − δ ) , (3.4b)with Q i , i ∈ { , . . . , } , being functions of ( x, y ). If Q = Q = Q = Q = δ = 1 and Q = Q = 0, the line element (3.2) reduces to that of Schwarzschild-AdS black hole inglobal coordinates, with y being related to the usual radial coordinate via r = L y + / (1 − y ),and x parametrising the standard polar angle on S with the identification sin θ = 1 − x .The reference metric is chosen to have Q = ε and, similarly to the soliton case,we work on a unit square grid. Regularity at x = 1 demands ∂ x Q i ( x, y ) = 0 ( i (cid:54) = 4), Q ( x, y ) = 0, and Q ( x, y ) = Q ( x, y ). At the conformal boundary, we set Q i ( x,
1) = 1 for i = 1 , , , Q ( x,
1) = 0 and Q ( x,
1) = ε .In the ansatz (3.4a) we have three parameters: ε , which sets the amplitude of theboundary rotation, and ( y + , δ ), which set the black hole temperature. The Hawking tem-perature is computed in the usual way by requiring smoothness of the Euclidean spacetimeand is given by T = (1 − δ ) y + δ (cid:0) y + 1 (cid:1) πy + . (3.5)For δ = 1, the temperature has a minimum at y + = 1 / √
3, coinciding with the minimaltemperature of a Schwarzschild-AdS , occurring at T Schwmin ≡ √ / (2 π ). In general we willhave two branches of solutions with the same temperature which we will refer to as large and small black holes. However, when ε (cid:54) = 0, our new deformed black holes can havea minimum temperature below that of the Schwarzschild-AdS black hole. In order tobypass this, we introduced a parameter δ which allows us to get arbitrarily close to zero– 6 –emperature . Computations with T > T
Schwmin were performed using the ansatz where δ = 1, and the results with δ (cid:54) = 1 enjoy similarly sufficient convergence properties as thosewith δ = 1 (see Appendix A). For simplicity of presentation, in all of our plots we will set L = 1. One of the most interesting results of our manuscript is that we find no soliton solutionsfor ε > ε c (cid:39) . ε > ε c .Furthermore, there is a range ε ∈ (2 , ε c ) in which two solitons exist for each value of ε ,demonstrating non-uniqueness within the soliton family.We have monitored the behaviour of the maximum of the Kretschmann scalar as afunction of ε . Since we have no matter fields, besides a negative cosmological constant,the Kretschmann curvature invariant K = R abcd R abcd is related to the norm of the Weyltensor in a simple manner L W abcd W abcd = L K − , that is to say, the local tidal forces scale like K / . Figure 2 : The log-linear plot of the maximum of the Kretschmann scalar against therotation parameter for the soliton solutions. The dashed gridline marks ε c , and the dottedline shows the max K = 24 /L for pure AdS . This ansatz allows us to reach zero temperature, for instance, see [33], where it was used to constructhairy AdS black holes, which in the T → We cannot exclude the possibility that more than two solutions might exist at some values of ε as weapproach ε = 2. – 7 – - - Figure 3 : Hyperbolic embedding of the cross section of the black hole horizons, for severalvalues of the parameter ε and a fixed temperature T = 1 /π . As we increase ε , the armsof the horizon cross section get stretched further apart. This picture is qualitatively thesame for T ≥ /π .The Kretschmann scalar K is maximal at the equator x = 0, and slowly deforms fromthe pure AdS value of 24 /L as we increase ε → ε c , forming two large extrema. Theminima grows only slightly slower than the maxima, and the latter is plotted against ε in Fig 2. For ε >
2, we can see that two soliton solutions exist for a fixed value of ε .We call the upper branch the large branch, and the lower branch the small branch. Thelarge branch of solitons smoothly extends the growth of K , which increases without bound,indicating formation of a curvature singularity. Also we note that for the soliton, the metriccomponent g tt tends to zero at the origin, possibly as ε → In order to obtain a better understanding of how the black hole horizon behaves under theincrease of the boundary rotation amplitude we analyze horizon embedding diagrams. Aswith Schwarzschild-AdS black holes, for a given temperature (3.5) there exist two solutionswhich we will call small and large black holes. In the case of the small black hole branch,we could successfully embed horizons into Eucliden space E for all ε - they are small andround spheres, slightly squashed through the poles. However, for the large branch, thescalar Gaussian curvature of the horizon spatial sections becomes too negative and theprocedure is only successful for very small ε . Instead, we embed spatial cross sections ofthe horizon into hyperbolic H space [34] in global coordinatesd s H = d R R / ˜ (cid:96) + R (cid:34) d X − X + (1 − X ) d φ (cid:35) , (4.1)– 8 – - - - - - - Figure 4 : Hyperbolic embedding of the cross section of the black hole solution horizons,for several values of the parameter ε and a fixed low temperature T = 0 . Left : Embeddings for the upper,thermodynamically dominant branch of solutions with ε ∈ (1 . , . ε = 2 . , . T = 1 /π , ε = 0 black hole. Right : Embeddings for the two small black hole branches. The higherentropy branch has ε ∈ (1 . , . ε = 2 . , . T regime.where ˜ (cid:96) is the radius of the hyperbolic space. For sufficiently small values of ˜ (cid:96) the embeddingalways exists. The induced metric on the intersection of the horizon with a partial Cauchysurface of constant t of the black hole line element (3.4a) is given byd s H = L (cid:34) y Q ( x, − x d x + y (1 − x ) Q ( x,
0) d φ (cid:35) . (4.2)The embedding is given by a parametric curve { R ( x ) , X ( x ) } . The pull back of (4.1) to thiscurve induces a two-dimensional line element with the following formd s = R (cid:48) ( x ) R ( x ) ˜ (cid:96) + R ( x ) X (cid:48) ( x ) − X ( x ) d x + R ( x ) (1 − X ( x ) ) d φ . (4.3)Equating this line element with (4.2) gives the following first order differential equation forthe polar coordinate0 = 4 H ( x ) P ( x )( X ( x ) − (cid:104) P ( x ) − ˜ (cid:96) ( X ( x ) − (cid:105) +4˜ (cid:96) P ( x ) X ( x )( X ( x ) − P (cid:48) ( x ) X (cid:48) ( x ) − ( X ( x ) − ˜ (cid:96) P (cid:48) ( x ) − P ( x ) (˜ (cid:96) + P ( x )) X (cid:48) ( x ) (4.4)with H ( x ) = (2 − x ) − (4 y Q ( x, P ( x ) = y (1 − x ) Q ( x, ε are presented in Fig. 3 where we set ˜ (cid:96) = 0 .
73. Here we are– 9 –xing the black hole temperature to be T = 1 /π , and the results for other temperaturesare qualitatively similar (Fig. 4).For ε <
1, the large black holes are round and the deformation is small. As we ap-proach ε → − , the black hole horizon gets significantly distorted and develops remarkableextended features. These cross-section “spikes” peak at x = (cid:113) − √ / i.e. θ = π/ . The horizons are squashed through the poles,and extended sideways, with the circumference slowly increasing. The black hole diameterthrough the poles tends to some non-zero value , and for very large rotation parameter thethickness of the cross section arms is surprisingly uniform throughout its length, and as afunction of ε . As we increase the temperature, the horizon deforms faster. We conjecturethat for sufficiently high temperatures, these static, axially symmetric solutions exist forany ε <
2, endlessly stretching the black hole horizon towards the AdS boundary. Thisis difficult to check numerically, but our results indicate that there is no upper bound tohow extended these black holes can become as we approach ε → − .Colder black holes turn out to be less elastic, and more extended through the equator.In Fig. 4 we plot embeddings for a low temperature T = 0 . ε wefind two or four black holes. Even though this phase is inelastic, as ε → + , we expect thestretched solutions to enter a scaling regime as with the hot solutions. As we tune the boundary rotation parameter, thus “stirring” the bulk, the black holebecomes ever more stretched and we expect the curvature on the horizon to increase cor-respondingly.In Fig. 5, we show the maximum of the Kretschmann scalar as a function of ε forlarge black holes. For the black holes, the Kretschmann scalar is maximal on the horizon,located at y = 0, and as ε → − , it peaks closer to θ = π/
4. For Schwarzschild-AdS , L K | H = 12 /y + 24 /y + 36, and thus for T = 1 /π the maximal value is 72 /L . As theboundary amplitude approaches ε = 2 − , the Kretschmann scalar grows without bound (seeFig. 5) and we expect the maximum of the Kretschmann scalar to enter a scaling regime.This quantity is difficult to extract accurately, however we find some numerical evidencethat it behaves as a power law as ε → − .For completeness, we present plots of the Kretschmann scalar of the thermodynamicallydominant solutions in Appendix B, Fig. 22. The entropy associated with the black hole horizon is given by S = A G N = 2 π y L G N (cid:90) d x − x √ − x (cid:112) Q ( x, Q ( x, . (4.5)In Fig. 6, we present our results for the small and large branches of black hole solutionswith a temperature T = 1 /π . For the large black holes, we find that the entropy is always As measured by the conformal boundary metric. This feature is not very apparent in the embedding diagrams. – 10 – .0 0.5 1.0 1.5 2.075100125150175
Figure 5 : Log-linear plot of the curvature scalar maximum against ε , for a large black holewith T = 1 /π . The dashed gridline marks the critical value of ε , and the dotted gridlineshows K max = 72 for a Schwarzschild-AdS black hole. The inset is a log-log plot for highvalues of the rotation amplitude. Figure 6 : Left : Entropy against the rotation parameter ε for the large branch of blackholes with T = 1 /π . The inset zooms around the limit, demonstrating the asymptoticbehavior. Red dashed gridlines mark the ε = 2 limit. Middle : Logarithmic derivative of S for the large black hole with T = 1 /π . As ε →
2, the entropy of the large black hole blowsup as an inverse power law.
Right : Entropy of the small branch of black holes with thesame temperature. Red dashed gridlines show where ε = 2 and S = 0.an increasing function of ε , and as ε → − , it grows without bound. When ε is sufficientlyclose to 2, the entropy enters a scaling regime as we saw with the Kretschmann scalar, and S ∝ (2 − ε ) − α (see the middle panel of Fig. 6). Numerically we find that α is consistentwith being 1 / Numerically, this is difficult to explore for temperatures much larger, and smaller than T = 1 /π . Thehot, large black holes reach the limit ε = 2 slowly, and the cold black holes are more difficult to resolve.However, we would expect that near the critical boundary value parameter ε = 2, the scalar invariants,such as entropy and curvature blow up according to some scaling laws. – 11 – .0 0.5 1.0 1.5 2.0 2.50.00.51.01.5 Figure 7 : Left : Entropy against the boundary parameter for low temperature black holeswith
T < T
Schwmin (cid:39) . T (cid:39) T Schwmin . Thebrown squares show the (hot) small black holes with T = 1 /π for reference. As we lowerthe temperature, the large black holes begin to exist for ε > Right : The asymptoticbehavior of isothermal entropy curves. Both solution branches approach ε = 2, with thelarge branch having a steeply increasing S , and the small branch having S →
0. The redgridlines show ε = 2 and ε = ε c .For the small branch (see right panel of Fig. 6), and fixed temperature, the entropy isa decreasing function of ε . In this case we can reach values ε >
2, although we still find amaximum value of ε beyond which we cannot find axially symmetric black hole solutions.We call this maximum value ε max ( T ). As the temperature decreases, ε max ( T ) approaches ε c from below, which is a turning point for the small branch of solutions, below which weobtain a second set of unstable solutions with lower entropy. In this region of the solutionspace numerical errors are increasingly large, and we were unable to extend the solutionsby further decreasing ε . We conjecture that S → ε → + , and the singularitymight be approached in a non-trivial way .This picture looks qualitatively the same for black holes with T > T
Schwmin (recall than T Schwmin (cid:39) . T . However, we can find black hole solutions with a minimal temperature T min ( ε ) < T Schwmin . This implies that for fixed
T < T
Schwmin , there are no black hole solutionsfor ε < ε min . As we cross the T Schwmin , the two black hole branches at fixed temperaturejoin up (see orange disks in Fig. 7), and as we lower the temperature further, the “left”turning point occurs for higher values of ε . The subsequent evolution of the phase space issomewhat intricate. It turns out, that for all fixed T < T
Schwmin , there exist large black holeswith ε >
2. There, we find yet another turning point, which bends back towards ε = 2.For temperatures T (cid:46) . ε = 2, and black holes exist In the literature, there have been examples where black hole solutions approaching a singular point ina phase space exhibit a non-trivial non-uniqueness, for instance in a spiralling manner (see e.g. [15, 33]). – 12 –nly for ε >
2. Soon after, once we reach T (cid:39) . T > T
Schwmin , we have two black hole solutions: for ε <
2, there is a large and asmall black hole, and for ε > ε → + as S →
0. For
T < T
Schwmin , we have two solutions for ε min ( T ) < ε < ε > T (cid:39) . ε max ( T ) < ε c .The low temperature regime is hard to access numerically, so we can only conjecture thevery low temperature behaviour. The only remaining turning point is decreasing slowlywith the temperature, separating the two branches of solutions the lower of which hasvanishing horizon area as ε → + . These solutions look qualitatively very similar tothe soliton solutions, and could be thought of as a small black hole placed in a solitonbackground. The upper branch has increasing entropy, which blows up as ε → + .In the following subsections we present physical quantities and carry out an exten-sive thermodynamic analysis of both black holes and horizonless AdS solutions, furthersupporting our conjectures. The bare on-shell gravity action in AdS is divergent and in order to obtain the bound-ary stress tensor we need to regularise the action using holographic renormalization tech-niques [35–37]. The counterterms that we must add to the bulk action (2.1) are S ct = − πG N L (cid:90) ∂ M d x √− γ (cid:32) − L R [ γ ] (cid:33) , (4.6)and the expectation value of the stress-energy tensor of the dual field theory can be foundto be (cid:104) ˜ T (cid:105) µν = 18 πG (cid:18) Θ µν − Θ γ µν + LG µν [ γ ] − L γ µν (cid:19) , (4.7)from the on-shell action.As an exercise, following [38], we also compute the holographic stress tensor by recast-ing the metric (3.4a) into Fefferman-Graham coordinates. Firstly, we solve the equationsof motion about the conformal boundary y = 1, order by order, by expanding the metricfunctions as shown below Q i ( x, y ) = (cid:88) n =0 (1 − y ) n q i,n ( x ) + log (1 − y )(1 − y ) τ i, ( x ) + O (cid:16) (1 − y ) (cid:17) . (4.8)Next we perform an asymptotic coordinate change near the AdS boundary to theFefferman-Graham form d s = 1 z (cid:104) d z + d s ∂ + z d s + z d s + O ( z ) (cid:105) . (4.9) Here and further in this subsection we set G N L = 1. – 13 – .0 0.5 1.0 1.5 - - - - - - - - Figure 8 : The stress-energy tensor component (cid:104) T tt (cid:105) ( left ), component (cid:104) T tφ (cid:105) ( middle ), andcomponent (cid:104) T θθ (cid:105) ( right ) for solitonic solutions as a function of θ , the angle on the S .Colours represent different values of the parameter ε , with larger values corresponding tolarger amplitudes.To achieve this, we take y = 1 + (cid:88) n =1 z n Y n ( θ ) + O ( z ) ,x = √ − sin θ + (cid:88) n =1 z n X n ( θ ) + O ( z ) , (4.10)and determine { Y n ( θ ) , X n ( θ ) } order by order in z so that our line elements can be cast inthe form (4.11) with conformal boundary metricd s ∂ = − d t + d θ + sin θ [d φ + d t cos θ ε ] . (4.11)The holographic stress-energy tensor is then given by (cid:104) T ij (cid:105) = 316 π g (2) ij , (4.12)where g (2) ij is the metric associated with d s . As expected, this stress-energy tensor istraceless and transverse with respect to (4.11). The energy density is simply given by ρ ( θ ) = −(cid:104) T tt (cid:105) , whilst the angular momentum density is simply j ( θ ) = (cid:104) T tφ (cid:105) . The independent boundary stress-energy tensor components for the soliton solution aregiven by (cid:104) T tt (cid:105) = 1768 π (cid:34) q ( θ ) + ε sin θ (cid:26) − ε + sin θ (cid:16) ε sin θ + 46 ε + 18 q ( θ ) (cid:17) − q ( θ ) (cid:27)(cid:35) , (4.13) (cid:104) T tφ (cid:105) = 1128 π sin θ cos θ (cid:104) ε sin θ + 3 ε − q ( θ ) (cid:105) , (4.14) (cid:104) T θθ (cid:105) = 1768 π (cid:104) ε sin θ (16 cos 2 θ −
27) + 18 q ( θ ) (cid:105) . (4.15)– 14 – .0 0.5 1.0 1.5 - - - - - - - - - - - - - - - Figure 9 : Independent stress-energy tensor components of the dual to the large black holesagainst θ , for several values of the parameter ε . Colouring codes black hole temperaturewith darker colours representing temperatures closer to T min ( ε ).Numerical results are presented in Fig. 8 where we plot these components against θ forseveral values of the boundary rotation amplitude ε . The stress-energy tensor component (cid:104) T tt (cid:105) is positive at the equator where it is also maximal; it becomes negative at a valuewhich depends on ε , and is negative at the poles. As the rotation amplitude increasestowards ε c , (cid:104) T tt (cid:105) develops a large negative minima in the upper quarter of the sphere. (cid:104) T tφ (cid:105) , on the other hand, is negative for small values of ε , and is vanishing on both thepoles and the equator. As ε → ε c , (cid:104) T tφ (cid:105) develops a broad, large maxima close to theequator, and for the second soliton branch, it becomes entirely positive. The component (cid:104) T θθ (cid:105) , measuring the pressure along θ , is positive and peaked at the equator for small valuesof ε . As the amplitude is increased, the pressure becomes very negative at the equator,increases towards the poles and develops a maxima close to θ = π/ .3.2 Black Holes The independent holographic stress-energy tensor components for the black hole met-ric (3.4a) are given by (cid:104) T tt (cid:105) = y + π (cid:20) y (cid:16) δ − δ − δ + 384 q ( θ ) − (cid:17) − (cid:16) δ − δ + 17 (cid:17) +16 ε sin θ (cid:40) sin θ (cid:18) − ε cos 4 θ + 72 q ( θ ) y + ε (cid:16) − δ + 12 ε − y + 2285 (cid:17)(cid:19) − q ( θ ) y + 32 ε (cid:16) δ + 9 y − (cid:17)(cid:27) (cid:21) , (4.16) (cid:104) T tφ (cid:105) = y + π sin θ cos θ (cid:20) ε sin θ − q ( θ ) y + 12 ε (cid:16) δ + y − (cid:17)(cid:21) , (4.17) (cid:104) T θθ (cid:105) = y + π (cid:20) (cid:16) δ + 26 δ − (cid:17) + 3 y (cid:16) − δ + 315 δ + 721 δ + 384 q ( θ ) + 97 (cid:17) − δ + 139) ε sin θ − ε sin θ (cid:105) . (4.18)Numerical results for several values of the black hole temperature and boundary rotationamplitude are presented in Fig. 9 (and for low temperatures see Appendix B, Fig. 24). Forsmall ε and large T , (cid:104) T tt (cid:105) is entirely negative. It is maximal at the equator and minimal ata value which depends on both ε and T , where it also attains its largest absolute value. Ifwe fix T and increase ε , the energy density at the equator decreases, and becomes slightlynegative. As expected, for a fixed boundary amplitude, the energy density is an increasingfunction of temperature.The component (cid:104) T tφ (cid:105) is negative for small ε and temperatures close to T min ( ε ), andvanishes at the poles and at the equator, similarly to the soliton case. Generally, (cid:104) T tφ (cid:105) isa complicated function of T and ε . For a fixed value of ε , it is an increasing function oftemperature, and for large enough temperatures it becomes large and entirely positive. Inparticular, when ε →
2, the density develops a large positive maxima peaked at θ = π/ θ is positive for all ε and T , and for a fixed rotationamplitude is an increasing function of temperature. Furthermore, for a fixed temperature,it is an increasing function of ε . In the large temperature, large ε regime, (cid:104) T θθ (cid:105) develops abroad positive maxima, peaked in the vicinity of θ = π/
4. The normal stress componentin the direction of φ (not pictured) can be negative around the equator, and in the large ε , T regime, it is positive with a sharp dip to negative values at θ = π/ We obtain conserved charges in the usual way by integrating the angular momentum andenergy densities [35]. We note that by construction, ε = 0 solutions take the Schwarzschild-AdS values, T = (1 + 3 y ) / (4 πy + ), S = πy , M = y + (1 + y ) / G = y + (1 − y ) / .0 0.5 1.0 1.5 2.0 2.50.00.10.20.30.40.50.6 Figure 10 : Left : Soliton solution boundary energy against rotation parameter. Thedashed vertical gridline marks the critical amplitude ε = 2, and the bold orange line isthe perturbative result. Right : Log-linear plot of the energy for the black hole solutionsagainst ε , for several values of T . Figure 11 : Left : Black hole entropy against temperature for several values of the boundaryrotation parameter ε , for ε <
2. The upper branches correspond to the large solutions andthe lower branches to the small solutions.
Right : Black hole energy as a function oftemperature. Colouring codes the value of the boundary amplitude. The bold black curveis the Schwarzschild-AdS solution with ε = 0.As expected, the total angular momentum is identically zero. We present the energy for black holes as a function of ε and T on the right panels of Fig. 10 and Fig. 11, re- We note that strictly speaking the quantity that we are computing is not an energy, since ∂ t is changingas we change (cid:15) . A better defined quantity would be ∆ E ≡ E black hole − E soliton , which could have beenregarded as the energy above the vacuum for each value of ε . – 17 – .25 0.30 0.35 0.40 0.45 0.50 - - - - - - - - - Figure 12 : Left : Gibbs free energy for the black holes against temperature for a fixedboundary rotation parameter ε = 1 (red data points). The orange gridline shows the valuefor the corresponding AdS soliton with the same ε . The grey solid curve traces the valuesof Schwarzschild-AdS , and the grey-dashed gridline shows G AdS = 0.
Middle : Similarplot for ε = 1 .
5, with the horizontal gridline marking the intersection temperature T HP ( ε ).To the right of this line the large black holes will be the dominant solutions, whilst solitonsdominate to the left. Right : As we approach the critical value of ε , the phase transitiontemperature decreases.spectively. For a fixed ε and varying T , the large black holes increase their energy, andthe small black holes decrease it, asymptotically approaching E →
0. Further, for a fixed T and increasing ε , the large black hole energy increases monotonically, as does the smallblack hole energy for ε <
2, which is the opposite behaviour to that of the entropy (see leftpanel of Fig. 11).For values 2 < ε < ε max ( T ), where max( T ) was identified in the discussion of Fig. 6and Fig. 7, the solution space has extra phases associated with cold temperatures. As welower the temperature, the black hole energy approaches ε = 2 from above, but unlikeentropy, it appears to blow up for both small and large solution branches. As we lower T further, black holes eventually exist only for ε >
2, and the two branches appear to havesimilar values of energy.The holographic energy for the soliton is presented on the left panel of Fig. 10. Withincreasing boundary parameter ε , the energy is increasing until it peaks at ε (cid:39) .
29 withmaximum energy E max (cid:39) .
527 for the small branch. The large branch has larger energy,which increases steeply when ε → + . We constructed these solutions numerically up to ε = 2 .
3, however numerical errors in this region of the parameter space become increasinglylarge.
Because we can fix the boundary chemical potential and temperature T , the appropriatethermodynamic ensemble for analysing our solutions is the grand-canonical ensemble. In– 18 – .6 1.8 2.0 2.2 2.40.350.400.450.500.550.600.65 1.6 1.8 2.0 2.2 2.40.350.400.450.500.550.600.65 1.6 1.8 2.0 2.2 2.40.350.400.450.500.550.600.65 Figure 13 : Gibbs free energy against the boundary parameter for low temepreature blackholes. It exhibits a swallowtail phase transition between the inelastic black hole phase andthe soliton (brown stars).this ensemble the preferred phase minimises the Gibbs free energy G = E − T S . (4.19)Note that in this system the energy, E , absorbs the chemical potential term, and in particu-lar for solitons we have G = E . This is further supported by the first law of thermodynamicsfor these solutions, derived following [39–41], and given byd E = T d S + 4 π (cid:90) π/ dΩ( θ ) j ( θ ) sin θ d θ, (4.20)where Ω( θ ) = ε cos θ is the boundary angular velocity density, and j ( θ ) is the angularmomentum density. In order to check the first law we constructed soliton solutions per-turbatively to second order, the procedure for which is detailed in Appendix C. We usethe second order expansion to calculate the boundary free energy to be E = 8 ε / (15 π ),which matches our numerical solutions to small ε amplitude (see left panel of Fig. 10) andsatisfies the first law (4.20) to first order in ε . Numerically, the first law is satisfied to atleast to 0 .
1% for the small soliton branch, to 2% for the large branch, and to 0 .
1% for theblack holes (with 2% for the second set of the small black holes), while varying ε . For blackholes it is also satisfied to 0 . G for theblack hole solutions as a function of the temperature for some values of the rotation am-plitude. For a fixed ε , the corresponding soliton energy (see Fig. 10) is constant, which ispictured as a line. The intersection of the two marks the generalised Hawking-Page tran-sition [21] , and the transition temperature T HP ( ε ) is a decreasing function of ε . Fortemperatures higher than the transition value, the large black hole phase is the dominantone, whilst for T < T HP ( ε ) the soliton is the preferred phase. Large black holes, and small black holes up to the turning point in the parameter space. The Hawking-Page transition occurs for ε = 0 and T = 1 /π (see the left panel of Fig. 12). – 19 – .0 0.5 1.0 1.5 2.0 2.50.220.240.260.280.300.320.34 Figure 14 : Phase diagram for the rotationally polarised solutions. For ε <
2, black holesexist above T min ( ε ) (black data points) and are the dominant phase above the T HP ( ε ) curve(red area) which marks the corresponding Hawking-Page transition. Below the transitionline (red data points), the dominant phase is the soliton. For ε >
2, there is a cold inelasticblack hole phase (red wedge). The elsewhere subdominant black holes exist up to somemaximum ε max ( T ), but are hidden by the soliton phase.For T < T
SchwHP = 1 /π , we observe curious phase transitions in G , characterised by a“swallowtail” behaviour in the Gibbs free energy [42, 43] (Fig. 13). This occurs becauselarge black holes start to exist past ε = 2, where there is a turning point in the parameterspace resulting in a four-fold non-uniqueness, and thus there exists a moduli space forwhich we have four possible black hole phases (and two soliton phases). We find that for0 . < T < /π , there is a temperature dependent interval ε HP ( T ) < ε < ε (cid:63) ( T ) with ε (cid:63) ( T ) >
2, where the cold, inelastic black holes dominate the ensemble. Finally, we notethat the small black hole branch is never favoured thermodynamically, as it always hashigher free energy than both the large black hole and the soliton. The correspondingphase diagram is illustrated in Fig. 14. We also present G as a function of ε for a fixedtemperature in Appendix B (see the left panel of Fig. 23). To investigate the stability, we consider a wave equation for a neutral, minimally coupledscalar field (cid:3)
Φ = 0 . (5.1)Since both solitons and black holes have two commuting Killing fields, ∂ t and ∂ φ , we candecompose the perturbations as followsΦ = (cid:98) Φ ω,m ( x, y ) e − i ω t + i m φ . – 20 – ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ��� ��� ��� ��� ��� ����������� ε ω ● � = � ■ � = � ◆ � = � ▲ � = �� ▼ � = �� ○ � = ��
20 30 40 502.22.32.42.5
15 20 25 30 35 40 45 500.0500.0750.1000.1250.150
Figure 15 : Left:
Normal mode frequencies with (cid:96) = m vs the boundary rotation parame-ter ε for the soliton metric. At ε = 0 these reduce to the usual AdS frequencies L ω = 3+ (cid:96) .Dashed gridlines show where ε = 2 and Re ω = 0. Right:
Rotation amplitude ε × , where ω becomes negative, against m . The inset shows the logarithmic derivative of ε × . We expectthat in the limit m → ∞ , ε × → (cid:98) Φ ω,m is linear and can be regarded as a Sturm-Liouville problemfor the eigenpair { (cid:98) Φ ω,m , ω } , once suitable boundary conditions are imposed. The set ofall ω are called the quasinormal mode (QNM) frequencies. For each value of m , thereis an infinite number of QNM frequencies labelled by two integers. These are in one toone correspondence with the number of nodes along the x and y directions. If ε = 0, wecan use the background spherical symmetry to decompose the angular part in terms ofspherical harmonics. These are labeled by (cid:96) and m , with (cid:96) − | m | counting the number ofnodes along the polar angular direction. We will be interested in following the modes with (cid:96) = | m | as a function of ε , that is to say, modes with no nodes along the polar direction.Furthermore, we will also be focusing on the modes which have no nodes along the radialdirection, i.e. with zero overtone. We focus on these modes since they have the largest | Im( ω ) | for fixed m .We follow the method of [16, 17, 31] and solve the resulting eigenvalue equation sys-tem using a Newton-Raphson method, on a fixed numerical solution background with acorresponding numerical grid.The QNM spectrum with ε = 0 is well–known: the Schwarzschild AdS black holeQNM frequencies are given in [44], and the scalar normal mode frequencies of AdS aregiven by integers L ω = 3 + (cid:96) . As we vary the boundary rotation parameter, the imaginarypart of the black hole frequency ω becomes positive, indicating the presence of an instability.The boundary conditions imposed at y = 0 will depend strongly on the backgroundsolution we wish to perturb. – 21 – .1.1 Soliton We first investigate scalar perturbations (5.1) on soliton backgrounds, and consider thefollowing decompositionΦ( t, x, y, φ ) = e − iωt e imφ y | m | (1 − y ) (1 − x ) | m | ψ ( x, y ) , (5.2)where the factorisation ensures regularity at the origin for a smooth function ψ ( x, y ). Theboundary conditions at y = 1 and x = 1 are obtained by solving the equation (5.1) at thecorresponding boundaries and are given by | m | ψ ( x, y ) + ∂ y ψ ( x, y ) = 0 at y = 1 , (5.3a)and ∂ x ψ ( x, y ) = 0 at x = 1 . (5.3b)We also require ∂ x ψ ( x, y ) = 0 at x = − ∂ y ψ ( x, y ) = 0 at y = 0.On the left panel of Fig. 15, we plot the normal mode frequencies ω with (cid:96) = m ,against ε . The ω first becomes negative when m = 13, and with each subsequent mode, ω becomes negative at a lower value of ε . We denote the value at which ω changes signby ε × , and we find that ε × − ∼ /m at large m (see right panel of Fig. 15).In order to better resolve where ω becomes negative for a given m , we can also set ω = 0 and look for zero–modes directly, and the results for this approach are presented inFig. 15, right. We find that this method gives us the same values of ε × as the ones inferredfrom computing ω directly. For each pair of values of { m, ε × } , we expect a given hairyfamily with nontrivial Φ to condensate. Whether this new family of hairy solutions willextend to large or small values of ε , is likely to depend on the model. We have constructedsoliton solutions with complex scalar hair within cohomogeneity two, and will present thesesolutions elsewhere. These were built by minimally coupling a massless complex scalar fieldto gravity, and are similar in spirit to the black holes with a single Killing field of [15] andto the holographic Q-lattices of [45]. Next, we consider the black hole solutions. A Frobenius analysis at the integration bound-aries motivates the following separation ansatzΦ( t, x, y, φ ) = e − iωt e imφ y − i ωy +1+3 y (1 − y ) (1 − x ) | m | ψ ( x, y ) , (5.4)where we have implicitly imposed regularity in ingoing Eddington-Finkelstein coordinates [44].The boundary conditions for ψ ( x, y ) at y = 1 and x = 1 are then found by solving(5.1) and demanding that ψ ( x, y ) has a regular Taylor expansion at y = 1 and x = 1. Thisin turn gives: − i y + ωψ ( x, y ) + (1 + 3 y ) ∂ y ψ ( x, y ) = 0 at y = 1 , (5.5a) Higher modes are more difficult to obtain numerically, and thus are less reliable, however, we find thatthe scalar eigenfunctions for these solutions have good convergence properties. For instance, we observe thevariation with the grid size in a mode with m = 40, ε = 2 . − . – 22 – .0 0.5 1.0 1.5 2.0 2.505101520
20 30 40 502.22.32.42.5
15 20 25 30 35 40 45 500.0500.0750.1000.1250.150
Figure 16 : Left:
Real part of quasinormal mode frequency with (cid:96) = m against theboundary rotation parameter ε , for small black holes with T = 1 /π . At ε = 0, these reduceto the Schwarzschild-AdS values. The bold dashed gridline shows the critical ε = 2 value. Right:
Rotation amplitude where the zero–mode is for a given m , against m . The insetshows the logarithmic derivative of ε × . We expect that in the limit m → ∞ , ε × → ∂ x ψ ( x, y ) = 0 at x = 1 . (5.5b)As with the soliton, we require ∂ x ψ ( x, y ) = 0 at x = − ∂ y ψ ( x, y ) = 0 at y = 0.Numerical results are presented in Fig. 16. On the left panel, we plot the real part ofthe QNM frequencies with (cid:96) = m against ε , for a fixed T = 1 /π small black hole branch.We find that the behaviour of Re ω is very similar to that of the soliton, and the firstunstable mode appears at m = 13. The imaginary part, Im ω , becomes positive exactlywhere Re ω crosses the zero axis, indicating a zero–mode. If we set ω = 0 and look forthese directly, we again find a 1 /m fall–off (see right panel of Fig. 16) at large m . Thegrowth rate of the instability is less than Γ ∼ − , so it sets in incredibly slowly . Weexpect that these solutions will become unstable in modes with high m as we increase theboundary rotation past ε = 2 and, if we assume the 1 /m decay, the instability shouldmanifest itself slowly with increasing ε . Finally, for hot, large black holes we do not seeany sign of instability, as these solutions never cross ε = 2.This is a surprising result - the ergoregion appears to induce hairy solutions in bothphases. This opens up new possibilities for the phase diagram, some of which we willdiscuss in the following sections. This section was devoted entirely to the stability analysisof scalar perturbations, and the issue of the stability of our backgrounds with respect togravitational perturbations is an open problem. However, if we assume that the physics We believe that the rate will increase for low temperatures, but these backgrounds require very highresolutions. – 23 –nvolved in this instability is any similar to the usual superradiance instability, we expectthe gravitational perturbations to be unstable and to have larger instability growth rates.
Our calculation in this section follows mutatis mutandis that of Section 4.1 of [46]. We willfirst consider a conformally coupled scalar field satisfying the Klein-Gordon equation on R t × S . The idea is to determine the eigenfunctions on this background, and use those todetermine the spectrum of a conformally coupled scalar on the rotating background (2.2).If ε = 0, the Klein-Gordon equation for a conformally coupled scalar in 1 + 2 reads(where we set the radius of the S to unity): − ∂ ∂t ϕ + ∇ S ϕ = 14 ϕ . The Hamiltonian eigenstates are given by spherical harmonics | (cid:96), m (cid:105) , with wave-functions (cid:104) θ, φ | (cid:96), m (cid:105) = Y (cid:96)m ( θ, φ ) . In this basis, the Hamiltonian ˆ H = i∂/∂t has the following spectrumˆ H | (cid:96), m (cid:105) = (cid:18) (cid:96) + 12 (cid:19) | (cid:96), m (cid:105) . After turning on the electric field, the Hamiltonian is deformed into H = (cid:114) − ∇ S + ε
16 sin θ + i ε cos θ ∂∂φ . Since H commutes with ∂/∂φ , it will remain diagonal in the azimuthal quantum number m .However, it is no longer diagonal in (cid:96) , and its elements have to be computed numerically: (cid:104) (cid:96) (cid:48) m (cid:48) | H | (cid:96)m (cid:105) = δ m,m (cid:48) (cid:40) (cid:90) π d ϕ (cid:90) π d θ sin θ ¯ Y (cid:96) (cid:48) m (cid:115) (cid:96) (cid:18) (cid:96) + 12 (cid:19) + ε
16 sin θ Y (cid:96)m − m ε δ (cid:96),(cid:96) (cid:48) − (cid:115) ( (cid:96) + 1 + m )( (cid:96) + 1 − m )(2 (cid:96) + 1)(2 (cid:96) + 3) + ( (cid:96) ↔ (cid:96) (cid:48) ) (cid:41) . (6.1)For each value of m , we can truncate the values of (cid:96) and (cid:96) (cid:48) up to some maximumvalue (cid:96) max and determine the eigenvalues numerically. One can show that if we are onlyinterested in a few low lying modes, the convergence is exponential in (cid:96) max . The resultsare illustrated in Fig. 17 where we plot the lowest lying eigenvalue of (cid:104) (cid:96) (cid:48) m | H | (cid:96)m (cid:105) , whichwe denote by λ m, , for several values of m = 2 , , ,
14. The similarity between the leftpanel of Fig. 15 and the left panel of Fig. 17 is striking. Note that Spherical harmonics are normalised such that (cid:82) π d φ (cid:82) π d θ sin θ | Y (cid:96)m ( θ, φ ) | = 1, for all (cid:96) and | m | ≤ – 24 – ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ��� ��� ��� ��� ��� ��� ��� - �� ����� ε λ � � � m ● ■ ◆ ▲ ▼ ○ ● ● ● ● ● ● � �� �� ����������������������� � ε � Figure 17 : Left:
Lowest lying mode eigenvalue of the boundary hamiltonian as a functionof the boundary rotation parameter ε . At ε = 0 these reduce to (cid:96) + 1 / Right: ε abovewhich λ m, becomes negative, plotted as a function of m . The solid line represents ouranalytic approximation (6.2) valid at large m .One can go further and compute the critical value of ε above which λ m, becomesnegative, and compare it with the right panel of Fig. 15. We call this special value ε ε × ,in accordance to section 5. In the large m limit, the Hamiltonian can be diagonalisedusing a WKB approximation, and can be used to determine ε × . We have performed thiscalculation to order m − , and it turns out that ε × = 2 + √ m − m + 132 √ m − m − √ m − m − √ m − m − √ m − m − √ m − m + O ( m − ) . (6.2)The validity of this approximation is tested on the right panel of Fig. 17, where we can seethat for m ≥
10, the error is below 0 .
1% and for m ≥
22 it is below 0 . The spacetimes we consider are stationary, however, the black hole solution has zero angularvelocity on the horizon, and vanishing total angular momentum. If we place a small Kerrhole on the axis of symmetry, there will be a general relativistic “spin–spin” force acting onthe spinning black hole, which for stationary fields can be thought of as analogous to theelectromagnetic “dipole–dipole” force with opposite sign [47–49]. In this section, we willinvestigate whether it is possible to have a spinning test particle hovering above the centralblack hole, balanced by the pull towards the black hole and the “spin–spin” interactionwith the boundary. – 25 –quilibria of spinning test particles can be studied using the Mathisson-Papapetrou(MP) equations [50–56] (for a review see [57]), which are derived by taking a multipoleexpansion about a suitably defined reference worldline. Truncated at dipole order, theyare given by Dp µ Ds = − R µνρσ v ν S ρσ ,DS µν Ds = p µ v ν − p ν v ν , (7.1)where p µ is the four-momentum of the particle, which in general is not co-linear with thekinematic four-velocity v µ . They are related by p µ = mu µ − u ν ˙ S µν , where S µν is the anti-symmetric spin matrix and m is a mass parameter . We also require a supplementary spincondition to fix the center of mass, which we choose to be the Tulczyjew-Dixon condition p ν S µν = 0. On the rotation axis, it is equivalent to the Mathisson-Pirani condition.We will be concerned with the static case, thus setting ˙ S µν = 0 and ˙ p µ = 0. Thisimplies m = M and p µ = M u µ , where M is the rest mass of the particle − M = p µ p µ . To-gether with the spin magnitude, defined as S = S µν S µν /
2, these are constant throughoutthe motion. These conditions, in conjunction with the MP equations (7.1), are sufficientto determine the static ratio
S/M as a function of the radial coordinate. Equivalently, wecan instead consider a conserved quantity which includes the spin potential energy E = − p ρ ξ ρ + 12 ξ ρ ; σ S ρσ , (7.2)where ξ = ∂ t is the time Killing vector field, and E > − M = p µ p µ , wecan find an effective potential for radial on-axis motion of a spinning particle. In order tofind equilibrium points, we look for local minima of the potential ˙ r + V ( r ) = 0, whichconstrains both the energy E , and the ratio S/M .We calculate these quantities for the black hole metric (3.4a) and the soliton met-ric (3.2). Because our coordinates are not well-defined on the symmetry axis θ = 0, wefirst perform a local coordinate change1 − x = x + x , tan φ = x /x , (7.3)and the symmetry axis is now given by x = 0 and x = 0. The supplementary conditionimplies that the only non-vanishing component of the spin matrix is S x x , making theon-axis calculation considerably easier. The effective potential is then found to be V ( r ) = ( g rr g tt ) − (cid:34) g tt + (cid:18) EM + SM y α Q (1 , y ) (cid:19) (cid:35) , (7.4) Here the dot is differentiation with respect to the proper time. This condition states that momentum is perpendicular to the covariant four-spin vector s = (cid:63) ( S ∧ p ). – 26 – .00.80.60.40.20.00.0 0.2 0.4 0.6 0.8 Figure 18 : Left : Spin to mass ratio for a spinning test particle in the soliton back-ground with boundary parameter ε = 0 . , . , , , . ε ≤
2) which, for small ε , are in a very good agreement with numerical solutions. Right : Spin to mass ratio for large black holes with T = 1 /π , for ε = 0 . , . , , . , . , . α = 1 for the soliton, and α = 2 for the black hole. The spin potential energy termdepends on the metric ansatz, and in the expression (7.4) it is evaluated explicitly. Thetotal energy of a spinning particle is EM = √− g tt − SM y α Q (1 , y ) , (7.5)and the spin to mass ratio of a static spinning test particle on the symmetry axis is SM = − ∂ y g tt (1 , y )2 y α − (cid:112) − g tt (1 , y )(2 α − Q (1 , y ) + y∂ y Q (1 , y )) . (7.6)Numerical results for both solutions are presented in Fig. 18. For the test body approxi-mation, the Møller radius describing the minimum size of a rotating body is R ≥ S/M [58],and it should be possible to make it arbitrarily small [47]. For the black holes, we do notfind such a limiting process. For the soliton, this quantity vanishes as y →
0, thus it ispossible to place a spinning particle at this point, whose backreaction would lead to theformation of a small Kerr-AdS black hole. For static spacetimes the equilibrium positions are independent of spin, and can be achievedby balancing electromagnetic and gravitational forces, or coincide with the static radius dueto a repulsive cosmological constant Λ >
0. For Kerr black holes [59], it is well known thatno equilibrium positions exist [47]. Generally, conditions for equilibrium on the rotation– 27 – .0 0.5 1.0 1.5 2.0 2.50.40.60.81.0
Figure 19 : Left : Minima of static potential, − g tt , for the soliton against the boundaryparameter. For ε <
2, there is a global minima at y = 0. For ε >
2, the minima is onlylocal, owing to the fact that there is an ergoregion attached to the boundary. For thesecond branch, the local minima is on the x = 0 axis, and for ε (cid:38) .
43, the minima isno longer local.
Right : Contour plot for − g tt for the second branch soliton, at ε = 2 . θ = π/ K ,whilst the shaded areas are the two minima with a steep maxima between them.axis are independent of the spin of the test particle. From (7.6), we can see that therequirement S/M → − g tt .The evolution of − g tt with the boundary parameter is non-trivial, and is related to thechanges in the Kretchmann scalar K . For the soliton with ε <
2, there is a global minimaat y = 0 (see the left panel of Fig. 19, and c.f. [60], Fig. 6). When ε ≥ − g tt becomes verynegative near the boundary due to the ergoregion, and the central minima is then local.For the second branch, the minima moves up the equator towards the boundary, and as ε →
2, the minima is found closer to where K has the largest gradient. The ergosurfacebecomes visibly deformed towards the x = 0 boundary, and the local minima eventuallyconnects to the ergoregion at ε (cid:38) .
43 (see the right panel of Fig. 19).For the large hot black hole solutions, the minima of − g tt is on the horizon, tending to θ = π/ ε →
2. The corresponding small branch has a turning point at ε >
2, where theentropy S → − g tt looks verysimilar to the soliton (see the right panel of Fig. 19), but with the local minimum alongthe horizon.Finally, we comment on the stability of the test particles. For the stable soliton branch– 28 – ε > r + , and temperature T , theentropy is given by S = 2 πr πr + T + 1 . (7.7)In the grand-canonical ensemble, at finite temperature T and fixed chemical potential µ ,we find that the change in the free energy is δG = M E + O ( r ), and thus adding (spinning)particles is not advantageous . In order to achieve a preferred phase, we would like tohave V min <
0, however, we find that the energy
E/M is positive . This is in agreementwith the fact that the small black hole phase is always hidden by the soliton phase.Once we cross ε = 2, any particle will eventually fall into the ergoregion near theboundary. Similarly, this should happen for the small black holes which exist past thecritical value ε = 2.It is interesting to contrast these results with a polarised black hole in global AdS [46, 61]. In these papers, solutions with a dipolar chemical potential on the boundary areconsidered. For the black holes, they find static charged particle orbits, which could allowfor multi-black hole solutions. In our scenario, we find that the purely gravitational on–axis“spin–spin” interaction is not sufficient to support stationary solutions with more than oneblack hole. The instabilities that we observe in the rotationally polarised AdS system are primarilydue to the induced ergoregion. It is interesting to ask, whether we still see instabilities ifwe isolate the boundary interaction term in the metric, in such a way that it does not causethe ergoregion to form. We consider the adjustment to the line elements (3.2) and (3.4a),such that g tt is always negative near the boundary, by choosing the reference metric in (3.1)to have Q ( x, y ) = 1 + ε x (1 − x ) (2 − x ) y . In this case, at any given fixed temperature T > T
Schwmin = √ / (2 π ), we also find a maximum rotation amplitude past which we do notfind stationary black hole solutions (see right panel of Fig. 20). As the system heats up,this value of ε appears to increase without a bound. We also find a soliton solution, whichappears to exist for any value of the rotation amplitude . We have been able to reach ε = 32, and believe that the solution exists for any value of ε . We have used the gauge/gravity duality to study the behaviour of a strongly coupledfield theory on a fixed rotating spacetime. Since our bulk field theory depends only on In the probe limit r + → The ratio is also independent of the spin sign, which agrees with the sign of S . The Kretschmann scalar exhibits a complicated structure, and for large ε has a slowly increasingcentral maximum, and two opposite sign extrema on the x = 0, approaching the boundary. – 29 – Figure 20 : Entropy vs the boundary rotation parameter, when the metric is adjustedso that there is no ergoregion on the boundary. We observe two branches of black holesolutions which join up at some maximum value of ε .the metric, it lies in the universal sector of AdS/CFT. In particular, it can be embeddedinto a full top-down model for AdS/CFT, such as ABJM [22]. We restricted attention toboundary spacetimes taking the following formd s ∂ = − d t + d θ + sin θ (d φ + ε cos θ d t ) , and studied the behaviour of several one point functions, such as the expectation value ofthe stress energy tensor, for several values of ε . We have considered both the vacuum stateand the thermal state for each value of ε . The holographic dual of the latter contains ablack hole solution in the interior of the bulk spacetime, setting the temperature of thefield theory [4].We have seen that both the thermal and the vacuum states are non-unique; for afixed angular boundary profile and fixed energy, more than one solution exists in thebulk. For boundary profiles containing ergoregions, we have observed a rather interestingbehaviour. Specifically, singular black hole solutions exist, extending from the bulk allthe way to the boundary, whenever the boundary profile has an evanescent ergosurface, i.e. when ∂/∂t becomes null along the equator at the boundary, but is everywhere elsetimelike. Furthermore, we have given ample numerical evidence that above a certain valueof ε = ε c (cid:39) . i.e. we determined the minimum valueof ε that makes our system develop hair, purely from CFT data.The most interesting question raised by our work does not yet have a definite answer.Namely, what happens if we start in the vacuum of the theory, i.e. pure global AdS , andpromote ε to be a function of time. In the most interesting scenario, we would increase ε to– 30 – value above ε c and monitor the bulk dynamical evolution. We have provided numericalevidence that there are no axially symmetric and stationary solutions above this criticalvalue. We have further seen that both the solitonic and the black hole phases can develophair for ε >
2. There are a number of possibilities for the nonlinear evolution of our system:1. The superradiant instability quickly controls the dynamics, and the subsequent evo-lution proceeds just as described in [15, 19, 20].2. A bulk horizon forms, becomes hot and expands all the way to the boundary, reachingthe boundary in a finite amount of time. This is reminiscent of what was observedin [62, 63].3. A bulk horizon forms, the curvature grows large in the bulk as a power law in time,until the superradiant instability kicks in and controls the dynamics. This scenariois very similar to the one presented in [64, 65].4. No horizon forms, and the curvature grows large in the bulk as a power law in time,until the superradiant instability kicks in and controls the dynamics. This scenariois again very similar to the one presented in [64, 65].We do not know which one of these possibilities is most likely, but were possibility 2 to bethe correct one, we would suspect that for our class of boundary metrics no positivity ofenergy theorem could be proven.There are several extensions of our work that deserve attention. First, it would beinteresting to consider a generalisation of this work to the Poincar´e patch of AdS . Workin this direction will be presented elsewhere. Second, it would be interesting to understandwhether the behaviour we observe depends on the rotation profile we choose. We thinkthat most of what we reported should not be dependent on the choice of the boundaryprofile. For instance, we certainly believe that no solution should exist for arbitrarily largeamplitudes of the boundary profile, so long as ∂ t becomes spacelike. Acknowledgments
JM is supported by an STFC studentship. JES was supported in part by STFC grantsPHY-1504541 and ST/P000681/1. It is a pleasure to thank J. Penedones for collabora-tion at an early stage of this work. The authors would like to thank C. R. V. Board and´O. J. C. Dias for reading an earlier version of this manuscript. JES would like to thankM. S. Costa, L. Greenspan and G. T. Horowitz for helpful discussions. The authors thank-fully acknowledge the computer resources, technical expertise and assistance provided byCENTRA/IST. Part of the computations were performed at the cluster Baltasar-Sete-S´oisand supported by the H2020 ERC Consolidator Grant “Matter and strong field gravity:New frontiers in Einstein’s theory” grant agreement no. MaGRaTh-646597. Part of thiswork was undertaken on the COSMOS Shared Memory system at DAMTP, Universityof Cambridge operated on behalf of the STFC DiRAC HPC Facility. This equipment isfunded by BIS National E-infrastructure capital grant ST/J005673/1 and STFC grantsST/H008586/1, ST/K00333X/1. – 31 –
Numerical validity
We verified that our solutions satisfy ξ µ = 0 to sufficient precision. In order to compute thestress energy tensor, adequate precision is required as we need to obtain third derivativesabout the conformal boundary. We found that double precision was adequate thus signif-icantly speeding up our computation. Nevertheless, we checked that the results convergeas expected when the precision is increased.We monitor convergence of our numerical method by computing the infinity norm ofthe DeTurck vector (cid:107) ξ (cid:107) ∞ , as well as relative errors of physical quantities of interest definedby ∆ N G = (cid:12)(cid:12) − G N +1 /G N (cid:12)(cid:12) , where G N denotes the quantity computed with N grid pointson each integration domain. This provides a good check of the numerics, as well as ameasure of the error. We present convergence results for free energy in Fig. 21 which, inthis work, is the physical quantity which exhibits the largest error. As ε increases, the gridsize has to be increased accordingly to maintain satisfactory error and typically a 60 ×
60 to100 ×
100 grid was used. For the dominant soliton branch, (cid:107) ξ (cid:107) ∞ is never above 10 − anderrors in quantities never above 0 . large branch, (cid:107) ξ (cid:107) ∞ < − , and the errorsare below 1%. For the large black holes, we keep (cid:107) ξ (cid:107) ∞ < − and ∆ N G < . small branch, 10 − and 1% respectively. Generally, we found that errors increase withboth increasing and decreasing the temperature.For analytic functions, we expect the pseudospectral methods to offer exponential con-vergence with an increasing grid size. However, we cannot attain these rates of convergenceif the error drops below the machine precision. The error settles to an exponential decayuntil it starts to pivot around the machine error, and this interval is shown in Fig. 21. Inorder to improve convergence it is then necessary to increase numerical precision. B Additional figures
In Fig. 22, we present the Kretschmann scalar K for a fixed value of the boundary parameter ε (cid:39) ε c , where ε c is the maximal value up to which stationary solutions exist. The evolutionof K with the boundary deformation is non-trivial. In the left panel, we plot K for the smallsoliton branch. For ε < .
9, the scalar has a positive minima at y = 0, and for ε > .
9, itstarts to display interesting features: the scalar is maximal at the equator x = 0, displayingtwo large extrema close to each other, the magnitude of which are growing rapidly withthe increase of the boundary rotation amplitude. At the boundary y = 1, it reduces to thevalue of K AdS = 24, as expected for the asymptotically AdS spacetimes. We also observea local slowly increasing maxima at y = 0, when ε is large. The large, unstable, solitonbranch shows a similar behavior, with magnitudes of the both extrema growing furtherand the distance between them decreasing, and the extrema are slowly moving towards theorigin, as ε →
2. There are also now two minima on each side of the maximum.In the right panel we present the curvature scalar for the large black hole with ε = 1 . T = 1 /π . It is maximal on the black hole horizon y = 1, and is peaking at θ → π/ ε →
2. On either side of the maximum there are two minima one of which is global.– 32 – - - - - -
40 45 50 55 60 -
30 35 40 45 50 55 6010 - - - -
65 70 75 80 85 - Figure 21 : Top left : Log–linear plot of the DeTurck norm squared for the dominatingsoliton branch, at ε = 2 .
5. Straight line indicates exponential convergence.
Top right :Log–linear plot of the relative error in the free energy G vs the grid size n , for the samedata set as top left. Bottom left:
Log–linear plot of the DeTurck norm squared vs gridsize for a large black hole with T = 1 /π and ε = 1 . Bottom right:
Log–linear plot of∆ N G against n , for a cold ( T = 0 . ε = 2 . Figure 22 : Left : Kretschmann scalar K for the dominant soliton branch, ε = 2 . Right : K for a large black hole with ε = 1 . ε sufficiently large, the K is small and positive on the poles and the equator.– 33 – .0 0.5 1.0 1.5 2.0 2.5 - - × × Figure 23 : Left : Gibbs free energy against the boundary profile for black holes with afixed temperature T = 1 /π (black disks), and soliton (brown squares). The dashed gridlinemarks the ε → Right : Maximum of the Kretschmann invariant against ε for thesmall black hole with a fixed temperature. As max K increases, the K looks very similarto that of the soliton. - - - - - - - Figure 24 : Boundary energy-stress tensor components for a fixed low temperature( T = 0 . C Perturbative expansion
In this section we present the perturbative expansion of the solitonic solution to order ε ,where ε is the boundary rotation parameter defined in section 2. The following procedurecan be easily generalized to yield higher orders of the expansion, however the solutionsbecome increasingly complicated. We will work in null quasi-spherical gauge [66] in whichthe solution readsd s = − (cid:32) r L (cid:33) Q ( r, θ ) d t + (cid:32) r L (cid:33) − Q ( r, θ ) d r + r Q ( r, θ ) (cid:104) d θ + sin θ (cid:0) d φ + Ω( r, θ ) d t (cid:1) (cid:105) (C.1)– 34 –here { θ, φ } are the standard polar coordinates of unit two-sphere, and L is the AdS length. The gauge requires that the functions Q i ( i = 1 , ,
3) and Ω depend only on r and θ , and we will also use the fact that if Q i = Q i ( r ) (and Ω = 0), we can additionally fix Q = 1. As the induced stress-energy tensor is of order ε in the angular velocity, consideran expansion in ε given by Q i ( r, θ ) = + ∞ (cid:88) i =0 ε i q (2 i ) i ( r, θ ) , Ω( r, θ ) = + ∞ (cid:88) i =0 ε i +1 Ω (2 i +1) ( r, θ ) (C.2)where we are expanding around a pure AdS background given by q (0) i = 1. At linear orderwe obtain a second order partial differential equation for Ω (1) which can be solved usingseparation of variables subject to regularity at the origin. The solution is given byΩ (1) ( r, θ ) = 23 √ π + ∞ (cid:88) l =0 c (1) l Γ (cid:16) l +32 (cid:17) Γ (cid:16) l +52 (cid:17) Γ (cid:16) l + (cid:17) P (cid:48) l +1 (cos θ ) (cid:18) rL (cid:19) l F (cid:32) l , l + 32 ; l + 52 ; − r L (cid:33) , (C.3)where l is the harmonic number, c (1) l are real numbers depending on the boundary profile, P l is Legendre polynomial of degree l and F is ordinary hypergeometric function. At theboundary the perturbation reduces tolim r → + ∞ Ω (1) ( r, θ ) = + ∞ (cid:88) l =0 c (1) l P (cid:48) l +1 (cos θ ) , (C.4)and at first order the first harmonic l = 1 gives us the dipolar boundary rotation profileΩ( r, θ ) = ε cos θ . The final expression isΩ (1) ( r, θ ) = 2 L πr cos θ rL (cid:32) r L + 3 (cid:33) + (cid:32) r L − (cid:33) (cid:32) r L + 1 (cid:33) arctan (cid:18) rL (cid:19) . (C.5)The first order angular velocity perturbation effectively sources the stress energy tensorat ε . We decompose the second order perturbations as [67] q (2) i ( r, θ ) = γ i ( r ) + α i ( r ) P (cos θ ) + β i ( r ) P (cos θ ) , (C.6)where the remaining gauge freedom is used to set γ ( r ) = 0. We solve the equations requir-ing that the functions are regular at the origin r = 0 and, that up to a diffeomorphism, theyasymptotically satisfy AdS boundary conditions. Below we provide the full perturbativefunctions: γ ( r ) = 2 L π r ( L + r ) (cid:18) L r + 20 L r − π − L r − (cid:16) L r + 7 L r − L r − Lr (cid:17) arctan (cid:18) rL (cid:19) + 4 (cid:16) L + 9 L r + L r + 7 L r + 9 r (cid:17) arctan (cid:18) rL (cid:19) − π r (cid:19) ,γ ( r ) = − L π r ( L + r ) (cid:32) (3 L − r )( L + r ) arctan (cid:18) rL (cid:19) − Lr (3 L + r ) (cid:33) – 35 – (cid:18) (3 L + r ) arctan (cid:18) rL (cid:19) − Lr (cid:19) ,α ( r ) = L π r ( L + r ) (cid:18) L r + L ( −
752 + 63 π ) r + 7 L ( −
16 + 15 π ) r + arctan (cid:18) rL (cid:19) (cid:16) − L r + 7 L (176 − π ) r + 2 L (368 − π ) r + L (80 − π ) r + 32( L + r ) (cid:16) L − L r + L r + 6 r (cid:17) arctan (cid:18) rL (cid:19) (cid:19)(cid:33) ,α ( r ) = L π r ( L + r ) (cid:32) L r + (4208 − π ) L r + arctan (cid:18) rL (cid:19) (cid:16) − L r +(63 π − L r + 2(63 π − L r + 32 L ( L + r ) (cid:16) L + 84 L r − r (cid:17) × arctan (cid:18) rL (cid:19) + (304 + 63 π ) r (cid:19) + (304 − π ) Lr (cid:33) ,α ( r ) = L π r (cid:32) − L r − π ) L r + arctan (cid:18) rL (cid:19) (cid:16) L r + (1360 + 63 π ) × L r − (cid:16) L + 19 L r − L r + 3 r (cid:17) arctan (cid:18) rL (cid:19) − (592 + 63 π ) Lr (cid:19) + 84 π r (cid:33) ,β ( r ) = L π r ( L + r ) (cid:32) π ) L r + 38(608 + 105 π ) L r + 21(224 + 81 π ) L r + arctan (cid:18) rL (cid:19) (cid:16) − π ) L r + 63(352 − π ) L r +3(7712 − π ) L r − L + r ) (cid:16) L + 144 L r + 48 L r + r (cid:17) arctan (cid:18) rL (cid:19) + (4448 − π ) Lr (cid:33) ,β ( r ) = L π r ( L + r ) (cid:32) − π ) L r − π ) L r + arctan (cid:18) rL (cid:19) (cid:18) π − L r + 15(315 π − L r +(2835 π − L r + 256 L ( L + r ) (cid:16) L + 189 L r + 61 r (cid:17) × arctan (cid:18) rL (cid:19) + 5(63 π − r (cid:19) − (4960 + 1701 π ) Lr (cid:33) ,β ( r ) = L π r (cid:16) − L (7072 + 735 π ) r + 7 L (544 − π ) r + 224 π r + arctan (cid:18) rL (cid:19) (cid:16) L (608 + 245 π ) r + 2 L ( − π ) r + L (2656 − π ) r + 128 (cid:16) L + 153 L r − L r − r (cid:17) arctan (cid:18) rL (cid:19) (cid:19)(cid:33) . (C.7)– 36 –sing this second order expansion we compute thermodynamic quantities, and test the firstlaw (see subsec. 4.4). We also find equilibrium conditions for spinning test particles on therotation axis θ = 0. For the metric (C.1), the spin to mass ratio (see sec. 7) is given by SM = πr εL √ r + 1 (cid:104)(cid:0) L + r (cid:1) tan − (cid:0) r/L (cid:1) − Lr (cid:105) + O ( ε ) , (C.8)and vanishes when r →
0. The comparison with the full non-linear numerical results arepresented in Fig. 18.
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